Prove that pivot columns of matrix A form a basis for C(A).
From "The Four Fundamental Subspaces".
Please show each step of your solution.
*Prove that pivot columns of matrix A form a basis for C(A).
(The material is from THE FOUR FUNDAMENTAL SUBSPACES. Please show each step of your solution. Thank you.)
Proof. First of all, we need the following facts
Fact 1: The dimensions of C(A) and R(A) are equal, since both are equal to the rank of A;
A matrix A is in Reduced Row-Echelon Form (RREF) if it has the following properties:
1) If a row does not consist entirely of zeroes, then the first nonzero entry in the row is a 1.
2) Any rows that consist entirely of zeroes are at the bottom of the matrix, below all nonzero rows.
3) For rows not consisting of all zeroes, ...
It is proven that pivot columns of matrix A form a basis for C(A).